Lectures on General Equilibrium Theory. Revealed preference tests of utility maximization and general equilibrium

Lectures ongeneral Equilibrium Theory Revealed preference tests of utility maximizationand general equilibrium p. 1/14 Lectures on General Equilibrium Theory Revealed preference tests of utility maximization and general equilibrium John Quah john.quah@economics.ox.ac.uk

Lectures ongeneral Equilibrium Theory Revealed preference tests of utility maximizationand general equilibrium p. 2/14 Observable Restrictions Let O = {(p t,x t )} t T be a set of observations. x t R l + is the observed demand at price p t R l ++. We know that certain observations are not consistent with utility-maximization with increasing utility functions. In other words, the utility-maximization hypothesis leads to observable restrictions on a data set. What restrictions on the data are sufficient for us to recover a utility function generating the data?

Lectures ongeneral Equilibrium Theory Revealed preference tests of utility maximizationand general equilibrium p. 3/14 Generalized Axiom of Revealed Preference The set O = {(p t,x t )} t T satisfies the generalized axiom of revealed preference (GARP) if whenever there are observations satisfying ( ): p t 1 x t 2 p t 1 x t 1 p t2 x t 3 p t 2 x t 2 p t n 1 x t n p t n 1 x t n 1 p t n x t 1 p t n x t n, then all the inequalities have to be equalities.

Lectures ongeneral Equilibrium Theory Revealed preference tests of utility maximizationand general equilibrium p. 4/14 Generalized Axiom of Revealed Preference Proposition: Suppose that O is drawn from a consumer with an increasing utility function. Then it must satisfy GARP. Proof: Suppose that O is drawn from a consumer maximizing the utility function U. If p s x r p s x s then U(x r ) U(x s ). And since U is increasing, if p s x r < p s x r then U(x r ) < U(x s ). The displayed inequalities in ( ) imply that U(x t1 ) U(x tn ) U(x tn 1 )... U(x t3 ) U(x t2 ) U(x t1 ). (2) Clearly we obtain a contradiction if any inequality is strict. So they must all be equalities, which mean that the observations satisfying ( ) must also be equalities. QED

Lectures ongeneral Equilibrium Theory Revealed preference tests of utility maximizationand general equilibrium p. 5/14 Afriat s Theorem Theorem: (Afriat) Let O = {p t,x t } t T be a set of observations. The following statements about O are equivalent: (1) There is an increasing and concave utility function U such that x s argmax x B(ps,p s x s ) U(x). (2) O obeys GARP. (3) There are numbers φ s and λ s, with λ s > 0 (associated to each observation s T ) such that φ s φ k +λ k p k (x s x k ) for any observations s and k. (4) The problem of finding (φ s,λ s ) (for each observation s) that solve (??) is a linear programming problem. Such problems are known to be solvable in the sense that there is an algorithm to determine in a finite number of steps, whether or not the system of linear inequalities given by (3) has a solution.

Lectures ongeneral Equilibrium Theory Revealed preference tests of utility maximizationand general equilibrium p. 6/14 Afriat s Theorem Whenever U is increasing, we know that O obeys GARP. Afriat s Theorem also say that if GARP holds, then there is a utility function generating O that is increasing and concave. Hence, if a data set is consistent with an increasing utility function, it must be consistent with an increasing and concave utility function. In fact, the utility function constructed from the data is U(x) = min (p t,x t ) O {φ t +λ t p t (x x t )}. This is always a concave function, and it is an increasing utility function since λ t > 0 for all t.

Lectures ongeneral Equilibrium Theory Revealed preference tests of utility maximizationand general equilibrium p. 7/14 Afriat s Theorem Proof of Afriat s Theorem: The first proposition says that (1) implies (2). Recall: (3) There are numbers φ s and λ s, with λ s > 0, such that φ s φ k +λ k p k (x s x k ) for any observations s and k. To see that (3) implies (1) consider U(x) = min (p t,x t ) O {φ t +λ t p t (x x t )}. This is an increasing utility function since λ t > 0 for all t. It also generates the observations in O. Let x satisfy p s x = p s x s, i.e., x is on the budget plane of the agent at price p s and income p s x s. It follows from the definition of U that U(x) φ s. To guarantee that x s maximizes utility in B(p s,p s x s ) it suffices to show that U(x s ) = φ s. This follows immediately from (3). QED

Lectures ongeneral Equilibrium Theory Revealed preference tests of utility maximizationand general equilibrium p. 8/14 Afriat s Theorem Proof that (2) implies (3): To simplify, assume that observations are generic, i.e., for any t t, p t x t p t x t. In other words, either p t x t < p t x t or p t x t > p t x t. Denote the set of observed demands by X, i.e., X = {x t } t T. The order on X is defined as follows: x t x s if p t x s < p t x t. In this case we say that x t is directly revealed preferred to x s. The order is the transitive closure of, i.e., x t x s if there are observations t 1, t 2,..., t n such that x t x t1 x t2... x tn x s. If x t x s we say that x t is revealed preferred to x s. We refer to as the revealed preference order on X derived from O.

Lectures ongeneral Equilibrium Theory Revealed preference tests of utility maximizationand general equilibrium p. 9/14 Afriat s Theorem Lemma: The order is transitive and irreflexive. An order on X is an extension of another order if x t x s whenever x t x s. Lemma: There is an extension of that is transitive, irreflexive, and complete, i.e., for any two elements x t and x s in X, either x t x s or x s x t. Note: There could be more than one such extension.

Lectures ongeneral Equilibrium Theory Revealed preference tests of utility maximizationand general equilibrium p. 10/14 Afriat s Theorem Lemma: Let be a complete extension of. Assume that X = {x 1,x 2,...,x N }, with x 1 x 2 x 3... x N. Then there are numbers φ s and λ s, with λ s > 0, such that φ s < φ k +λ k p k (x s x k ) for all k s Proof: Denote p i (x j x i ) by a ij. Choose φ 1 to be any number and λ 1 to be any positive number. Since x j x 1 for all j > 1, So there is φ 2 such that a 1j = p 1 (x j x 1 ) > 0. φ 1 < φ 2 < min j>1 {φ 1 +λ 1 a 1j }. Now choose λ 2 > 0 sufficiently small so that φ 1 < φ 2 +λ 2 a 21. Since a 2j = p 2 (x j x 2 ) > 0 for j > 2, we have φ 2 < min j>2 {φ 2 +λ 2 a 2j }. Therefore, we can choose φ 3 such that { } φ 2 < φ 3 < min min {φ 2 +λ 2 a 2j },min {φ 1 +λ 1 a 1j }. j>2 j>1

Lectures ongeneral Equilibrium Theory Revealed preference tests of utility maximizationand general equilibrium p. 11/14 Afriat s Theorem Therefore, we can choose φ 3 such that { } φ 2 < φ 3 < min min {φ 2 +λ 2 a 2j },min {φ 1 +λ 1 a 1j }. j>2 j>1 We can choose λ 3 > 0 and sufficiently small such that More generally, we can choose φ k such that φ i < φ 3 +λ 3 a 3i for i = 1,2. φ k 1 < φ k < min s k 1 and λ k > 0 and sufficiently small so that { min j>s {φ s +λ s a sj } φ i < φ k +λ k a ki for i k 1. } In this way, we obtain φ s < φ k +λ k a ks for all k s, as required. QED

Lectures ongeneral Equilibrium Theory Revealed preference tests of utility maximizationand general equilibrium p. 12/14 Brown-Matzkin Theorem Consider an observer who makes several observations of the outcome of an economy. Each observation consists of a price vector p t, aggregate endowment ω t, and the income distribution {w a t} a A. With no loss of generality, assume that a A wa t = p t ω t = 1. We denote the set of observations by O = {(p t,ω t,{w a t} a A )} t T. Note that individual demands are not observed.

Lectures ongeneral Equilibrium Theory Revealed preference tests of utility maximizationand general equilibrium p. 13/14 Brown-Matzkin Theorem Suppose that O = {(p t,ω t,{wt} a a A )} t T is Walrasian rationalizable, so we can find increasing utility functions U a, generating the demand function x a, such that x a (p t,wt) a = ω t for every observation t. a A Since individual endowments are not observed, each observation can be thought of as arising from an exchange economy where agent a has utility function U a and endowment w a t ω t. The data is generated by changes to endowments, while the utility function of each agent is assumed to be unchanged across observations.

Lectures ongeneral Equilibrium Theory Revealed preference tests of utility maximizationand general equilibrium p. 14/14 Brown-Matzkin Theorem Theorem: (Brown-Matzkin) Given any set of observations O, there is an algorithm that could determine in a finite number of steps whether or not the set of observations is Walrasian rationalizable. Proof: By Afriat s Theorem, O is Walrasian rationalizable if and only if there is x a t 0 such that (1) a A xa t = ω t for all t; (2) p t x a t = w a t for all a and t; and (3) the set {p t,x a t} t T obeys GARP for all a. By Afriat s Theorem again, this is equivalent to finding (for every t and a) x a t 0, φ a t, and λ a t > 0, such that (1) a A xa t = ω t for all t; (2) p t x a t = wt a for all a and t; and (3) φ a s φ a k +λa k p k (x a s x a k ) for all a, k and s. This is a system of quadratic inequalities. It is known that these are solvable (Tarski-Seidenberg Theorem). QED